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In this article we extend the Bloch-Wigner exact sequence over local rings, where their residue fields have more than nine elements. Moreover, we prove Van der Kallens theorem on the presentation of the second $K$-group of local rings such that their residue fields have more than four elements. Note that Van der Kallen proved this result when the residue fields have more than five elements. Although we prove our results over local rings, all our proofs also work over semilocal rings where all their residue fields have similar properties as the residue field of local rings.
In this article we prove a generalization of the Bloch-Wigner exact sequence over commutative rings with many units. When the ring is a domain, we get a generalization of Suslins Bloch-Wigner exact sequence over infinite fields. Our proof is differen
A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups $G$ for which the integral group ring $mathbb{Z}G$ has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condi
We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over $mathbb Z,$ and study connections between them.
Let $widetilde{X}$ be a smooth Riemannian manifold equipped with a proper, free, isometric and cocompact action of a discrete group $Gamma$. In this paper we prove that the analytic surgery exact sequence of Higson-Roe for $widetilde{X}$ is isomorphi
The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new ind